Basic Theory of SMC

2. Variable Structure Control
• Variable structure control (VSC) is a form
of discontinuous nonlinear control.
• The method alters the dynamics of
a nonlinear system by application of a high-
frequency switching control.
• The state-feedback control law
is not a continuous function of time.

3. Variable Structure Control –
Constituent Systems
Mode 1
Mode 2
1x a x 
2
2 10
x a x
a a
 
 

4. Piecing together

5. Properties of VSC
►Both constituent systems were oscillatory
and were not asymptotically stable.
►‘Combined’ system is asymptotically stable.
►Property not present in any of the constituent
system is obtained by VSC

6. Another Example – Unstable
Constituent Systems
0x x x   0x x x   

7. Analysis
►Both systems are unstable
►Only stable mode is one mode of system
►IF the following VSC is employed
2
0,
2 4
x x x
 
        
0
, * , *
0
I xs
Mode s c x x c
II xs


    


8. Combined

9. In this case
►Again, property not present in constituent systems
is found in the combined system.
►A stable structure can be obtain by varying
between two unstable structures.
►However, a more interesting behavior can be
observed if we use a different ‘switching’ logic.
2 2
0
, ,0 *
0
I xs
Mode s c x x c c
II xs

    


10. The Region

11. Sliding Mode
New trajectory that was not present in any of the
two original systems

12. Sliding mode?
►Defined : Motion of the system trajectory along a
‘chosen’ line/plane/surface of the state space.
►Sliding Mode Control : Control designed with the
aim to achieve sliding mode.
 Is usually of VSC type
Eg : Previous problem can be perceived as
0
sgn( )
x x u
u xs x


  
 

13. Design of Sliding Mode Control
 Phase 1 (Sliding Surface Design):
Constructing Switching Surfaces so that the
system restricted to the switching surface
produces a desired behavior.
 Phase 2 (Controller Design):
Constructing switched feedback gains which
drive the plant state trajectory to the sliding
surface and maintain it there.

14. Advantage of SMC
Robustness
Nonlinear Structures
Uses information about input as well as output
feedback in control determination.
It is an open-closed-loop type control.
Improve performance based on computed torque
Insensitive to the uncertainties variation

15. Required Property
►For sliding mode to be of any use, it should have
the following properties
System stability confined to sliding surface
(unstable sliding mode is NOT sliding mode at
all)
Sliding mode should not take ‘forever’ to start
Disadvantages:
Very large control effort

16. The Chattering Problem
►When, s is very close to zero, the control signal
switches between two structures.
►Theoretically, the switching causes zero
magnitude oscillations with infinite frequency in x.
►Practically, actuators cannot switch at infinite
frequency. So we have high frequency oscillations
of non-zero magnitude.
►This undesirable phenomenon is called chattering.

17. Response
Ideal Sliding Mode
Practical – With
Chattering

18. Why is chattering undesirable?
►The ‘high frequency’ of chattering actuates un-
modeled high frequency dynamics of the system.
Controller performance deteriorates.
►More seriously, high frequency oscillations can
cause mechanical wear in the system.

19. Chattering Avoidance/Reduction
►The chattering problem is because signum
function is used in control.
Control changes very abruptly near s=0.
Actuator tries to cope up leading to ‘maximum-
possible-frequency’ oscillations.
►Solution :
Replace signum term in control by ‘smoother’
choices’

20. Disadvantage of ‘smoothing’
►If saturation or tanh is used, then we can observe
that near s=0
►
►Where represents the saturation or tanh
function.
►The limit in both cases is zero.
►So, technically the sliding mode is lost
0
( )lim
s
s
kf s
s
 
( )f s

21. QUESTIONS